When we deal with the continuity equation, we assume that the porosity of porous media is constant (Theta=n.Sw) so we can take the "n" as a fixed value and our partial equation will have only Sw and q as variables.
Now, if we take the "n" value as variable (temporal variable), which assumptions would govern the equation? If the porosity can be assumed as a variable (in deformable soils), what will be the form of the continuity equation?
You can take the total time derivative of 1-n, which is the compacity
1-n=Vs/V
where
n : porosity
V : total volume
Vs : solid volume
Assuming no phase exchange in a saturated soil, the result
(dn/dt)/(1-n)=(dV/dt)/V-(dVs/dt)/Vs
See for example the book of O. Coussy, Mechanics of Porous Continua, Wiley, 1995.
The term (dV/dt)/V is the volume strain rate of the porous medium.
The term drs/dt is the solid volume variation. This variation may reflect the compressibility of the solid component (as in poromechanics). It may also account for internal erosion.
The case of unsaturated soil can be treated as well. See for example the book of D.G. Fredlund and H. Rahardjo, Soil Mechanics for Unsaturated Soils, Wiley, 1993.
You can take the total time derivative of 1-n, which is the compacity
1-n=Vs/V
where
n : porosity
V : total volume
Vs : solid volume
Assuming no phase exchange in a saturated soil, the result
(dn/dt)/(1-n)=(dV/dt)/V-(dVs/dt)/Vs
See for example the book of O. Coussy, Mechanics of Porous Continua, Wiley, 1995.
The term (dV/dt)/V is the volume strain rate of the porous medium.
The term drs/dt is the solid volume variation. This variation may reflect the compressibility of the solid component (as in poromechanics). It may also account for internal erosion.
The case of unsaturated soil can be treated as well. See for example the book of D.G. Fredlund and H. Rahardjo, Soil Mechanics for Unsaturated Soils, Wiley, 1993.
You could also read Smiles (2000), Hydrology of swelling soils: a review, Aust. J. Soil Res., 2000, 38, 501–21. which can be accessed via http://www.publish.csiro.au/?act=view_file&file_id=SR99098.pdf.
void ratio e, which is equal to porosity n by n/(1-n), was used in a recent framework to describe volumetric behaviour of compacted soils. e, ew (moisture ratio) and net stress are used in the framework and suction is replaced by moisture ration ew. This makes this framework much easier to apply in field.
Paper can be found at:
http://users.monash.edu.au/~kodikara/highlights.html
First you need to answer two questions: 1) changes in porosity due to changes in pressure or foreign (external) factors? 2) permeability will vary with changes in porosity? Remarque - porosity is not included in the flow continuity equation and a direct link between porosity and specific storage missing
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